Find an explicit formula for the geometric sequence $2\,,\,6\,,\,18\,,\,54,...$. Note: the first term should be $\textit{b(1)}$. $b(n)=$
Solution: In a geometric sequence, the ratio between successive terms is constant. This means that we can move from any term to the next one by multiplying by a constant value. Let's calculate this ratio over the first few terms: $\dfrac{54}{18}=\dfrac{18}{6}=\dfrac{6}{2}={3}$ We see that the constant ratio between successive terms is ${3}$. In other words, we can find any term by starting with the first term and multiplying by ${3}$ repeatedly until we get to the desired term. Let's look at the first few terms expressed as products: $n$ $1$ $2$ $3$ $4$ $f(n)$ ${2}\cdot\!{3}^{\,0}$ ${2}\cdot\!{3}^{\,1}$ ${2}\cdot\!{3}^{\,2}$ ${2}\cdot\!{3}^{\,3}$ We can see that every term is the product of the first term, ${2}$, and a power of the constant ratio, ${3}$. Note that this power is always one less than the term number $n$. This is because the first term is the product of itself and plainly $1$, which is like taking the constant ratio to the zeroth power. Thus, we arrive at the following explicit formula (Note that ${2}$ is the first term and ${3}$ is the constant ratio): $b(n)={2}\cdot{3}^{{\,n-1}}$ Note that this solution strategy results in this formula; however, an equally correct solution can be written in other equivalent forms as well.